Multilevel importance sampling for rare events associated with the McKean–Vlasov equation

Abstract This work combines multilevel Monte Carlo with importance sampling to estimate rare-event quantities that can be expressed as the expectation of a Lipschitz observable of the solution to a broad class of McKean–Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator introduced in this context in Ben Rached et al. (Stat Comput, 2024. https://doi.org/10.1007/s11222-024-10497-3 ) to the multilevel setting. We formulate a novel multilevel DLMC estimator and perform a comprehensive cost-error analysis yielding new and improved complexity results. Crucially, we devise an antithetic sampler to estimate level differences guaranteeing reduced computational complexity for the multilevel DLMC estimator compared with the single-level DLMC estimator. To address rare events, we apply the importance sampling scheme, obtained via stochastic optimal control in Ben Rached et al. (2024), over all levels of the multilevel DLMC estimator. Combining importance sampling and multilevel DLMC reduces computational complexity by one order and drastically reduces the associated constant compared to the single-level DLMC estimator without importance sampling. We illustrate the effectiveness of the proposed multilevel DLMC estimator on the Kuramoto model from statistical physics with Lipschitz observables, confirming the reduced complexity from $${\mathcal {O}(\textrm{TOL}_{\textrm{r}}^{-4})}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mtext>TOL</mml:mtext> <mml:mrow> <mml:mtext>r</mml:mtext> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for the single-level DLMC estimator to $${\mathcal {O}(\textrm{TOL}_{\textrm{r}}^{-3})}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mtext>TOL</mml:mtext> <mml:mrow> <mml:mtext>r</mml:mtext> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance $$\textrm{TOL}_{\textrm{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>TOL</mml:mtext> <mml:mtext>r</mml:mtext> </mml:msub> </mml:math> .